Comparisons —
Each of these notions of superfunctions leads to a particular different notion of supermanifolds, by gluing open subsets of \(\mathbb {R}^{m|k}_{L^{\prime }}\)  though \(H^\infty \) -, \(G^\infty \) - or \(GH^\infty \) -diffeomorphisms, see [1]}. Again most of the `physical' applications requires the use of \(G^\infty \) - or \(GH^\infty \) -supermanifolds. One can compare the various theories as follows:
